Archimedian Theorems for Composite Solids

TL;DR

This paper investigates the properties of the center of gravity in composite solids, deriving conditions for its lowest position and providing explicit calculations for various shapes, bridging mathematical and physical insights.

Contribution

It introduces new theoretical results on the center of gravity in partly filled solids and derives a differential equation governing its behavior.

Findings

Center of gravity is lowest when on the top surface of the material inside the solid.

Derived a differential equation for the first moments of the solid.

Explicit calculations for cylinders, cones, spheres, and other solids.

Abstract

We consider the Center of Gravity of a solid, partly filled with some homogeneous material, and find its qualitative and quantitative properties. In particular, we prove that the Center of Gravity has its lowest position when it lies on the top surface of the material inside the solid and find a differential equation for the first moments that explains this result in both mathematical and physics terms. We make explicit calculations of this lowest position in a number of cases, such as cylinders, cones, solids of revolution, power solids, spheres and half spheres.